BC 135 
.B6 
1914 
Copy 1 



PROBLEM 

SCIENCE=ANALYSIS 

FORMULA 

INDETERMINATE 

SCIENCE=ANALYSIS 

DETERMINATE 

BY 

GEORGE ASHTON BLACK, Ph.D. 



Man gewinnt dadurch schon sehr viel, wenn man eine 
Menge von Untersuchxingen unter die Pormel einer 
einzigen Aufgabe bringen kann. Denn dadurch erleich- 
tert man sicb niebt allein selbst seiii eigenes Geschaft. 
indeni man es sicb genan besthnmt, sondern auch jedem 
anderen, der es priif en will, das Urtheil, ob wir unserem 
Vorbabenein G-eniige gethan haben oder nicbt. — Kant. 



NEW YORK 

PEINTED FOE THE AUTHOR 

1914 



Copyright, 1914, by 
George Ashton Black 

Published April, 1914 



MAY -12 J9K 



THE DE VINNE PRESS 

©CI.A374049 



."B6 

l± \ + DEFINITION 

Science = cognition necessary and sufficient to re- 
solve all cases of a general problem = Analysis. 

PROBLEM 

Science = Analysis 

FORMULA 

Indeterminate 

Science = Analysis 

Determinate 

WORK 

Science = Analysis 

Science { predicate } Analysis 

a . f not any predicate | . , ; 

Science j any j^^e | Analysis 

NOTE 

Formula is a convenient notation of the logical division 
necessary and sufficient to determine the case of problem 
universally through different cases in deductive sequence. 
The case in which problem is required to be indeterminate 
is the singular case. The case in which problem is required 
to be determinate is the general case. 

The first moment of work is the only solution of the 
singular case. In semblance it is only problem itself. In 
truth it is no determination of problem in reference to the 
necessary correlate of any subject of discussion, namely, 
any predicate. The second moment of work, as any deter- 
mination of problem in reference to any predicate, is, in the 
order of deduction, the first of all possible solutions of the 
general case. The third moment of work, as the sequence of 
negative and affirmative determination of problem in refer- 
ence to any predicate, is the general solution to the form of 
which all solutions of the general case that are different and 
successive in reference to the first can be reduced. The 
sequence of second and third moments is necessary and 

3 



sufficient to solve the general case universally. The whole 
of work in giving the only solution of the singular case, 
and the only necessary and universal solution of the general 
case, gives the required resolution of all cases. 

PROBLEM 

Science { not any predicate } Analysis 

NOTE 

The only necessary resolution of this negative problem con- 
sists in the absence of formula and work in the case of the 
problem. After the next problem in the order of deduction 
has been resolved, namely problem Science { any predicate } 
Analysis, the resulting predicates can be made negative by 
connecting them with the word not; and then by means of 
these negative predicates a different resolution of the pres- 
ent problem can be produced, but without extending our 
knowledge of Science = Analysis in the least. For the refer- 
ence of not any predicate to any subject renders the reference 
of different negative predicates to the subject superfluous. 

PROBLEM 

Science { any predicate } Analysis 

FORMULA 

Science I ni S nest > that is most general, predicate 1 Anfllvsis 
Science | any inferior predicate | Analysis 

WORK 

Science {g^} Analysis 

according to classic definition of abstract in logic, of func- 
tion in mathematics. 

NOTE 

In logic classic distinction between abstract and applied 
so defines abstract that it cannot be any inferior predicate, 

4 



and can only be the highest predicate, of which any inferior 
predicate is some application. In mathematics classic defini- 
tion, for instance Dirichlet's, requires function to be some 
determination of a variable in reference to a rule. As in 
any case either negatively or affirmatively determined in 
reference to a rule, the denned function cannot be the singular 
highest predicate, and can only be the general any inferior 
predicate, in which one inferior predicate and another infe- 
rior predicate are connected. 

The single series o min. max. co 

belongs in abstract. The two connected series 

co max. min. o min. max. co 

belong under abstract, that is in function. One and the 
same degree, for instance zero, has a certain singular char- 
acter as proper to the single series, and a certain general 
character as common to the two connected series. In refer- 
ence to the rule either plus or minus ± , every degree proper 
to the single series is indeterminate, that is neither nega- 
tively nor amrmatively determined. In reference to the same 
rule any degree common to the two connected series is de- 
terminate, that is either negatively or amrmatively deter- 
mined; in particular zero as neither plus nor minus is 
negatively determined, and any other degree as either plus 
or minus is amrmatively determined. Implicit in the two 
connected series is the variety 



minimum 

co < maximum > ± 

[ min. and max. J 

where the empty place corresponding to the place filled by 
the actual sign of either plus or minus, is, in that corre- 
spondence, the potential sign of neither plus nor minus. 
Comparison and distinction of degrees in abstract belong in 
abstract. Comparison and distinction of degrees and signs 
in function belong in function. 

5 



Science 



Science 



limit 



singularity 



PROBLEM 

abstract 
FORMULA 

abstract 

mathematical 

metaphysical 

logical 

WORK 

o 

CO 

minimum 
maximum 



Analysis 



Analysis 



abstract 



mathematical 



e"lit y { ^feneZs } -taphysica! 

,.^ absolute ) , . , 

universahty j relative | logical 



APPLICATION OF WORK 



variable 



thing 



thought 



imag. 
real 

no 
some 



minimum 

maximum 

min. and max. 



individual 

general 

universal 

none 

one 

other 

one or other 



none 



any 



indistinct{ thought } immediate" 

f concept 1 

distinct J j udgment I mediate 

] syllogism! 

6 



function 



in general 



reference of 

cognition 

to an object 



NOTE 

Helpful to an understanding of the present work is a new 
use of the circle comparable to the classic use of the same 
by Euler in logic. Witness 

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First different and successive in reference to the whole of 
the unsigned use 

o min. 

© © 

oo max. 

is the corresponding whole of the universally signed use 

o min. =*= 

© © 

oo ± max. =*= 

in which the empty place in the case of zero corresponding to 
the place filled by the actual sign of either plus or minus in the 
case of any other degree, is, in that correspondence, the poten- 
tial sign of neither plus nor minus; and from which can be de- 
veloped, in the shape of the presented application, the sure and 
unchanging foundations of mathematics, metaphysics and 
logic as different sciences, not as reduced to the form in 
which they become different parts of science in the abstract. 

"Toute la science mathematique repose sur Hdee de 
fonction," remarked Emile Picard in a lecture given at 
Clark University in 1899. The primacy of thing in general 
in metaphysics was remarked by Kant in his transcendental 
analytic. The primacy of thought in logic defined to be 
science of the laws of thought was ever sufficiently in- 
dicated by the definition. 

Accordingly mathematics is able and bound to extend its 
sway first over its own domain, and then over the domains 
hitherto ruled by metaphysics and logic. Nor is the pre- 
scribed extension only prescribed. From beginning to end 
it has been achieved once and again at least ; once in the 
present work and again in the presented application. 

In the first two of the four principal parts of the presented 
application there is given or found the doctrine or demon- 
strated theory of function, through which and according to 
which any doctrine of different functions, for instance any 
formula of problem Science { function } Analysis, first be- 
comes possible. 

8 



The theory of functions in classic mathematics depends 
upon the two connected series of numbers 

oo ... 3 f 2 f l 1, 2, 3 ... oo 
where zero is neither plus nor minus, and any other num- 
ber is either plus or minus. Implicit in the two connected 
series of signed numbers is the variety 

0{0 i 

oo \ 2. 3 ... I ± 



CO 



variable 



function 



where the empty place corresponding to the place filled by 
the actual sign of either plus or minus, is, in that corre- 
spondence, the potential sign of neither plus nor minus. 
The comparison and distinction of the numbers and signs 
conspicuous in the variety is recorded in the same notation 
as before in the case of the corresponding variety of signed 
degrees on page 6. Therefore the notation expresses a com- 
mon theory of function in respect of which classic mathe- 
matics and the present treatise are identical, however differ- 
ent they are in other respects. 
The notation in question 

imag. { } 

I individual 

real -! general V x 

[ universal J 

exhibits, in necessary and universal reference to function as 
any inferior predicate, the sequence of one inferior predicate 
conveniently read blank, and another inferior predi- 
cate Xj where the sequence of and x represents any 
moment of the well-ordered logical determination of imag- 
inary and real in necessary and universal reference to varia- 
ble, apart from which, by definition, function is impossible. 
Since there is only one moment in the whole of the logical 
determination of imaginary, any moment of that determina- 
tion is properly represented by the notation of that one in 
the place corresponding to the place of x representing any 
moment of the logical determination of real. 

9 



In the metaphysical part of the presented application, 
terms definitive of increasing diversity in metaphysical gen- 
erality are compared and distinguished, the identity and 
difference in which were hitherto imperfectly discriminated. 
Witness remarks of Bertrand Russell, in his "Principles of 
Mathematics," on "any and kindred words." In the logical 
part, it was a needed innovation to posit the classic and 
universal solution of the general case of problem 

thought = reference of cognition 

to an object 

in dependence upon the new-found and only solution of the 
singular case. 

Remarkable a priori is the prevalence of the principle of 
equivalence everywhere in every conceivable science. It was 
first remarked, so far as I know, by August Ludowici of Ge- 
neva, Switzerland, in a letter to me dated 27 February, 1909. 
Interesting to every rational acquainted with Kant's theory 
whether of the different successive steps in the process of 
making any object by degrees completely known according to 
a constant rule, or of what constitutes perfection in any cog- 
nition referred to any object, is the corresponding practice 

f intuition synthesis ] 

ideal -i concept — analysis > method 
I idea = dialectic I 

where the single place in semblance empty is in truth filled 
as the sequel will show, and where the single straight line 
is the constant rule according to which the two different suc- 
cessive lines in the single symbol of equivalence are produced. 

The presence of two equivalent formulations of the theory 
is explained by the fact that Kant distinguishes the same 
cognitions, regarded from two different points of view, on 
the one hand as determinations of a known object, on the 
other hand as acts of a knowing subject. 

Use of classic mathematical expressions in a new sense 
which can be gathered from the new context with mathe- 
matical certainty and precision has been made, not pointed 

10 



out, hitherto j and will continue to be made, not pointed 
out, in the sequel. But in this place it is convenient to call 
attention, once for all, to the use in question, by point- 
ing to the example of it contained in the rest of this para- 
graph. The two equivalent formulations, when posited in 
one to one correspondence, and prior to any construction 
of them, perfectly differentiate one and the same whole of 
possible cognition; that is logically divide it into a series 
of empty compartments. The integration of the resulting 
differential equation, that is the filling of the said com- 
partments in their proper order and connection by the 
only cognitions necessary and sufficient to fill them, first 
reduces the formulated theory to practice ; first constructs 
it mathematically; first makes, according to that theory, the 
unknown cognitive function by degrees completely known. 
The first compartment is filled by seeing in it the cognition 
termed a plane in geometry, and there defined after the 
definition of a straight line; but here accepted as axiomatic 
groundwork of the postulated straight line — as first refer- 
ence of any regular superstructure to any regular ground- 
work, and used to begin the presented development of the 
cognitive function into a series. 

The construction of the two equivalent formulations of 
the theory raises a certain problem. For the resulting series 
and the equivalence of the two formulations differ as intui- 
tion and concept in respect of the same object, namely the 
theory. Therefore a corresponding idea remains to be in- 
vented or discovered. I find it in the necessary reference 
of the terms of the series as mathematical to the compari- 
son and distinction of them expressed in the language of 
mathematics. The resulting idea or dialectic 
"1 f axiom 

_ t = J postulate 

= J [ definition 
supplies all that was wanting to the perfection of the first 
complete knowledge of the theory according to the theory 
itself as a constant rule. 

11 



Science 



Science 
Science 



Science 
Science 



PROBLEM 

function 

FORMULA 

x 
WORK 



individual 

general 
universal 



Analysis 



Analysis 
Analysis 




according to the doctrine of function deduced on page 6 
and noted on pages 8, 9. 



A: 



POSTULATE 
I 



a plane equilateral triangle the internal determination 
of which is only imaginary; =the solution of one case of the 
general problem, Science { } Analysis. 

II 

a plane equilateral triangle the internal determina- 
tion of which is real in respect of only the middle point 
of the altitude; = the solution of one case of the general 
problem, Science {individual} Analysis. 

Ill 

Resolution of all cases of Science {individual} Analysis = 
solution of one case of Science {general} Analysis. 

IV 
Resolution of all cases of Science {general} Analysis = solu- 
tion of one case of Science {universal} Analysis. 

HENCE 

by corresponding stages: 
12 



I Resolution of all cases of Science { } Analysis achieved 
in and through the integration of a simple differential 
equation derived from the solution of one case. 



A 



intuition 



/ concept 

A 



idea 



ideal 



of Science 



Analysis 



/ synthesis 

^/ analysis 

/ \ dialectic 



method 



ideal 
of Science 



intuition 
concept 



idea 



A 



synthesis 
analysis 

dialectic 



method 
of Analysis 



where the only imaginary predicate, as a part of speech, is 
an adjective which is read blank not only in two places 
picked out by small braces, but also in two different places 
not so picked out. 

13 



II Resolution of all cases of Science { individual } Analysis 
achieved in and through the integration of a simple differ- 
ential equation derived from the solution of one case. 



intuition 
concept 
/•\ idea 

/• synthesis 

/ analysis 

/\ dialectic 



ideal 
of individual 
Science 



intuition 



y ideal 



of Science { individual } Analysis 



> method 



synthesis 



concept D analysis 

A B 



idea 



AB 
x 

CD dialectic 
II 
ABC-D 



method 

of individual 

Analysis 



14 



Ill Resolution of all cases of Science { general } Analysis 
achieved in and through the integration of a complex dif- 
ferential equation derived from the solution of one case. 

A 

c 

D 
A B 

AB 

x 

CD 

II 

ABC-D 



intuition -\ 
concept l ideal 
idea J 

of Science { general } Analysis 
synthesis -> 
analysis I method 
dialectic J 

method of general Analysis 
synthesis analysis dialectic 



© intuition 



m C z zZ 

% concept 



D D D D 

A B x y X Y xX yY 



^ AB base multiplicand base = area ■— y 2 alt. 

O XXX XXX 

^ idea CD %. altitude multiplier ^ altitude = area -j- base 
| II II II II II II 

22 ABC'D area product area = area 2 -;- area 

where the complex form of the differential equation in 3 is 
rendered necessary by the specialty of the particular Science 
\ x I Analysis in question. Only through the cross refer- 
ence of the two equivalent formulas obtained from 2 could 
the same whole of possible Science { general } Analysis be 
perfectly differentiated a priori. 

15 



IV a Derivation from the solution of one case, of a com- 
pound differential equation demanding the resolution of all 
cases, of Science { universal } Analysis. 



A A 



c 

D 
A B 

AB 

x 

CD 

II 

ABC'D 



D 



Z 

D 

X Y 



zZ 

D 

xX yY 



base multiplicand base = 

X X XXX 

Yi altitude multiplier y 2 altitude = area -i- base 



area 



product 



II II 

area = area 5 



area 



ideal 



single manifold restrictive intuition 

definite definable definitive concept 

individual general universal idea 

2 of Science { universal } Analysis 

synthetical analytical dialectical synthesis 

" " " analysis ^method 

" " " dialectic J 



3 Here belong the empty tables ABC which follow. 
They perfectly differentiate the whole of possible Science 
{ universal } Analysis, and constitute a compound differen- 
tial equation demanding the resolution of all cases of that 
problem. 



16 



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[ 



IV b Essay to refer recognized moments of the demanded 
resolution of all cases of Science { universal } Analysis each 
to its proper place in the formula demanding the resolution. 

On the supposition that pure reason, constant as the 
faculty of Science = Analysis in all rationals of all 
times, but varied through all moments of the more and 
more definite and particular use of that faculty according 
to the rule function in different rationals of different times, 
has somewhere in some context already cognized every step 
in the solution of every case of Science { universal { Analysis, 
but has not yet recognized any step in the solution of 
any case in its proper place in the resolution of all cases ; 
I propose to search out all and only the cognitions that are 
the content of that resolution, and arrange them each in that 
proper place as fixed for it a priori by the empty tables ABC. 
To be sure the task is not for only one rational, but for 
every one interested in the development as much as 
possible in himself of the same faculty that aforetime made 
the cognitions, and is now in his person called upon to 
recognize what it has itself in other persons already cognized 
according to a fixed and ascertained formula. As my own 
discovery of the required cognitions and reference of them 
to this or that place in the formula is sure and complete as 
regards the solution of at least the first or singular case of 
the general problem in question, so all that is wanting to 
the perfection of the demanded resolution of all cases will 
undoubtedly be found, if able men, and such as are ac- 
quainted with what is classic in the use of pure reason, will 
endeavor to recognize the missing cognitions by the general 
but sufficient marks that relegate them to one or another 
place in the formula in correlation with one or another mo- 
ment of the singular solution. 



20 



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22 



NOTE 

At the end of table B, in three conspicuous compartments, 
is laid down what was hitherto lacking, a mathematically 
well ordered curriculum of the sciences deduced from a 
mathematical definition of science. The unit of the curric- 
ulum is one group. In reference to the first group as pro- 
totypal, the eleven different successive groups are ectypal. 
The first group is classic mathematics, meaning arithmetic 
algebra geometry once as separate sciences, and again as 
combined and raised to a higher power in the calculus. The 
organon of the extension of our knowledge in respect of the 
first ectypal group is the present treatise. It deserves to 
be called posterior mathematics in reference to classic mathe- 
matics as prior. The next two groups it is convenient and 
exact to call dynamics, in analogy with Kant's separation of 
the categories connected with relation and modality as dy- 
namical, from the categories connected with quantity and 
quality as mathematical. It is also convenient and in con- 
formity with classic usage to call the first four groups phys- 
ics, the next four logic, the last four ethics. 

In prior mathematics there is no generally accepted notion 
of algebra corresponding to the notion of arithmetic as sci- 
ence of number. Nevertheless function is here required to 
be the subject-matter of algebra by the nature and position 
of the indicia discursive magnitude and plurality referred 
to quantity, corresponding to the derivation of function 
whether from signed number or from signed degree 5 and 
elsewhere function has been found to. be the subject-matter 
of algebra by at least one respectable mathematician, namely 
Auguste Comte. 

In logic the symbol of equivalence expresses a thought 
which as a concept is such that any different concept is 
subordinated to it according to the logical series 

predicate 
predicate {^ 

23 



LIBRARY OF CONGRESS 



020 196 927 A 

of which the logical equation 

predicate = predicate < n ^ 

is only a transformation. The same thought expressed in 
the series and again in the equation is yet again expressed 
in the remark that the concept equals is the identical ground 
of the complete determination of any different concept. 
In logic also the state of logical extent, in case every degree 
thereof is neither negatively nor affirmatively determined 
in reference to the rule ±, is sufficiently indicated by the 
notation, log. ex. unsigned. The state of the same, in case 
any degree thereof is either negatively or affirmatively de- 
termined in reference to the same rule, is sufficiently indi- 
cated by the notation {log. ex. } ± where the empty place 
to the left of the left-hand brace, corresponding to the place 
filled by the actual sign of either plus or minus, is, in that 
correspondence, the potential sign of neither plus nor minus. 
In ethics, according to the demonstration on pages 10, 11, 
of what constitutes perfection in any cognition referred to 
any object, the production by degrees of the symbol of 
equivalence 



beginning with the inspection of the cognition termed a 
plane in geometry, first gives reason in the person of the 
producer something to do that it can do in respect of every 
one of its faculties in their proper order and connection. 
Accordingly such a production of this symbol is the neces- 
sary and sufficient means to the end of first educating the 
whole of reason in that person to a certain extent. First 
education is here definitive of any further education ; for of 
course, in the order of deduction, only through first and 
according to first does any further ever become possible. 



24 



